Mesh not moving with Armature in pose mode
Regarding the first image, those vertices aren't weighted to any bone, you'll have to select the corresponding bones and weight paint them, and in the last image:
1 Put the main mesh in object mode
2 Select all the bones in pose mode, hold shift and click the main mesh, then click back on the bones while still holding shift
3 parent with automatic weights
4 reapply weights on the parts that don't deform as intended